liquid and the solid state are given by the corresponding values for the concentrations of the liquidus and solidus curves. The ratio of the atom numbers in the two phases obey the lever rule (2.19). With decreasing tem- perature the solidified fraction of the melt increases, and the Si-concentration in the crystalline phase decreases. It is therefore not possible for continuously miscible alloys to grow a crystal out of a melt that has a homogeneous con- centration ratio, unless one confines the crystallization to a small fraction of the melt.
One can utilize, however, the dierent equilibrium concentrations in the melt and the solid to purify a crystal of undesirable impurities. This is the basis of purification by zone melting: One begins by melting a narrow zone of a crystal rod at one end. In this molten zone the impurity concentration necessarily is as it was in the solid. Then, the molten zone is slowly pulled over the rod. If the liquidus and solidus curves are as in Fig. 2.14 with regard to an impurity (with Si as the base material and Ge as an impurity) then the re-crystallized rod in the cooling zone has a lower concentration of impurities than the (respective) melt. Hence, the impurities are enriched at that end of the rod that is molten last. A large section of the crystal can very eectively be purged of impurities by repeating the process many times.
distribute nint interstitial atoms onN'interstitial sites is N0!=nint! N0 nint!.
For a Frenkel pair the number of interstitial atoms nint necessarily equals the number of vacancies nv. With n=nint=nv one obtains for the entropy S(comp. 2.12)
S ln N!
n! N n! ln N0! n! N0 n!
NlnNN0lnN0 2nlnn N nln N n 2:22 N0 nln N0 n:
In equilibrium the system is in the state of lowest free energyF=nDE ±TS, in whichDEis the energy required to create a Frenkel pair. The correspond- ing equilibrium concentration hni is obtained by dierentiating the free energy with respect ton
dF
dn0DE Tln hni2
N0 hni N hni : 2:23
Since the concentration of defects is small hni N;N0 hni is approxi- mately
hni
N N0
p e DE=2 T: 2:24
Hence, the concentration rises exponentially with the temperature according to an Arrhenius law. The activation energy in the Arrhenius law is half the energy required for the creation of a Frenkel-pair. The factor of two in the denominator of the exponent arises because vacancies as well as interstitial atoms are distributed independently in the crystal. One therefore has two independent contributions to the entropy. If the atom that is displaced from the regular site diuses to the surface or into an interface (``Schottky defect''), the full energy of creation for the defect appears in the Arrhenius law. The reason is that for a macroscopic solid the number of available sites on the surface or in an interface are infinitely small compared to the number of sites in the bulk. In that case only the vacancies in the regular crystal sites contribute to the entropy.
Defects of the next higher dimension are line defects. A common intrin- sic line defect is the dislocation. A simple example is shown in Fig. 2.20, which displays a cross section of a crystal with a dislocation. Around the core of the dislocation atoms are displaced from their regular lattice posi- tions because of the elastic stresses. Most of the energy required to create a dislocation is actually in the elastic strain that decays rather slowly as one moves away from the core. Dislocations are described by theBurgers vector.
The Burgers vector is constructed by considering the positions of atoms after completing a closed loop of an arbitrary size around the dislocation core for a lattice with and without a dislocation (Fig. 2.20). If the Burgers vector is oriented perpendicular to the dislocation line as in Fig. 2.20, the
k k
k
k
k
dislocation is called an ``edge dislocation''. If the Burgers vector is oriented along the dislocation line, then the dislocation is called a ``screw disloca- tion'', since by moving along a closed loop around the dislocation one climbs from one lattice plane onto the next. Edge dislocations and screw dislocations represent two limiting cases of a general, intermediate form of a dislocation. Furthermore, the angle between the orientation of the Burgers vector and the dislocation line may vary as one moves along the dislocation line. The modulus of the Burgers vector is equal to the distance of an atom plane for the common screw or edge dislocations. However, dislocations for which the modulus of the Burgers vector is only a fraction of a distance between an atom plane also exist. Such a partial dislocationis generated, e.g., if all atoms in a section of an fcc-crystal are displaced along a direction in a densely packed plane so as to produce a stacking fault (Fig. 2.9).
Dislocations play a crucial role in plastic deformation of crystalline material. Consider a shear force acting parallel to an atom plane. It is not feasible to make all the atoms glide simultaneously since the shear force works against the atomic bonds of all atoms in the glide plane at once. A step wise glide is energetically much more favorable. Firstly an edge disloca- tion is generated at the surface and then the dislocation is shifted through the crystal. Then gliding is eectuated by displacing the atoms row-by-row until the dislocation line has moved through the entire crystal. The required forces are much lower since fewer atoms are aected and bonds need not be broken but must merely be strained and re-oriented. Plastic deformations of a crystalline solid are therefore connected with the generation and wander- ing of dislocations. In a pure, ideally crystalline material dislocations can move easily, provided the temperature is not too low. For many metals,
b
A B
Fig. 2.20.Sectional drawing of a crys- tal with an edge dislocation (sche- matic). The dashed line represents a loop around the core of the disloca- tion. The loop begins with atom A. It would close at atom B if the disloca- tion were not present. The Burgers vectorbpoints from atomAto atom B. The same Burgers vector is ob- tained for any arbitrary loop that en- closes the dislocation
e.g., room temperature suces. Such materials have little resistance to plastic deformation. Examples are rods or wires consisting of annealed cop- per and silver. If the material is polycrystalline, e.g., after cold working, then the wandering of dislocations is hindered by the grain boundaries between the crystallites, and the material resists plastic deformation more eectively.
Problems
2.1 The phase transition from graphite to diamond requires high pressure and high temperature in order to shift the equilibrium in favor of diamond and also to overcome the large activation barrier. Suggest a method of pro- ducing diamond (layers) without the use of high pressure.
2.2 Below 910°C iron exists in the bcc structure (a-Fe). Between 910°C and 1390°C it adopts the fcc structure (c-Fe). Assuming spherical atoms, deter- mine the shape and size of the octahedral interstitial sites in c-Fe (a= 3.64 AÊ) and ina-Fe (a= 2.87 AÊ). Sketch the lattices and the interstitial sites. For which phase would you expect the solubility of carbon to be high- er? (Hint: The covalent radius of carbon is 0.77 AÊ.) When molten iron con- taining a small amount of carbon (91%) is cooled, it separates into a more-or-less ordered phase containing a-Fe with a small concentration of carbon atoms on interstitial sites (ferrite) and a phase containing iron car- bide (cementite, Fe3C). Why does this occur? Why does cementite strength- en the medium against plastic deformation?
(Hint: Fe3C, like many carbides, is very hard and brittle.)
2.3 Copper and gold form a continuous solid solution with the copper and gold atoms statistically distributed on the sites of an fcc lattice. For what re- lative concentrations of copper and gold atoms do you expect ordered alloys and what would they look like? Draw the unit cells of these alloys and iden- tify the corresponding Bravais lattices. Can you suggest an experiment which would determine whether the alloy is ordered or not?
2.4 Draw and describe the symmetry elements of all Bravais lattices.
2.5 Draw the primitive unit cell of the fcc lattice and determine the lengths of the primitive lattice vectors a', b', c' (in units of the conventional lattice constant a) and also the angles a', b', c' between the primitive lattice vec- tors. (Hint: Express the primitive lattice vectors as a linear combination of the lattice vectors a, b, c of the face-centered cubic lattice and use elemen- tary vector algebra.) What distinguishes this unit cell from that of the rhom- bic Bravais lattice?
2.6 Determine the ratio of the lattice constants c and a for a hexagonal close packed crystal structure and compare this with the values ofc/afound for the following elements, all of which crystallize in the hcp structure:
He (c/a=1.633), Mg (1.623), Ti (1.586), Zn (1.861). What might explain the deviation from the ideal value?
2.7 Supposing the atoms to be rigid spheres, what fraction of space is filled by atoms in the primitive cubic, fcc, hcp, bcc, and diamond lattices?
2.8 Give a two-dimensional matrix representation of a 2-, 3-, 4-, and 6-fold rotation. Which representation is reducible?
2.9 Show that the rhombohedral translation lattice in Fig. 2.3 is equivalent to a hexagonal lattice with two atoms on the main diagonal at a height of c/3 and 2c/3. Hint: Consider the projection of a rhombohedral translation lattice along the main diagonal into the plane perpendicular to this main diagonal. The main diagonal of the rhombohedral lattice is parallel to the c-axis of the hexagonal lattice. How are the a and the c axis of the corre- sponding hexagonal lattice related to the angle a=b=c and the length a=b=cof the rhombohedral lattice?
2.10 Take a piece of copper wire and anneal it by using a torch! Demon- strate that the wire is easily plastically deformed. Then pull the wire hard and suddenly or work it cold using a hammer. How is the plastic behavior now? Explain the observations!
A direct imaging of atomic structures is nowadays possible using the high- resolution electron microscope, the field ion microscope, or the tunneling microscope. Nonetheless, when one wishes to determine an unknown struc- ture, or make exact measurements of structural parameters, it is necessary to rely on diraction experiments. The greater information content of such measurements lies in the fact that the diraction process is optimally sensi- tive to the periodic nature of the solid's atomic structure. Direct imaging techniques, on the other hand, are ideal for investigating point defects, dis- locations, and steps, and are also used to study surfaces and interfaces. In other words, they are particularly useful for studying features that represent a disruption of the periodicity.
For performing diraction experiments one can make use of X-rays, electrons, neutrons and atoms. These various probes dier widely with re- spect to their elastic and inelastic interaction with the solid, and hence their areas of application are also dierent. Atoms whose particle waves have a suitable wavelength for diraction experiments do not penetrate into the so- lid and are thus suitable for studying surfaces. The same applies, to a lesser extent, to electrons. Another important quantity which diers significantly for the various probes is the spatial extent of the scattering centers.
Neutrons, for example, scatter from the nuclei of the solid's atoms, whereas X-ray photons and electrons are scattered by the much larger (*104 times) electron shells. Despite this and other dierences, which will be treated in more detail in Sect. 3.7, it is possible to describe the essential features of dif- fraction in terms of a single general theory. Such a theory is not able, of course, to include dierences that arise from the polarization or spin polar- ization of the probes. The theory described in Sect. 3.1 below is quasi classi- cal since the scattering itself is treated classically. The quantum mechanical aspects are treated purely by describing the probe particles as waves. For more detailed treatments including features specific to the various types of radiation, the reader is referred to [3.1±3.3].